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1. The energy you give to the battery, and its mass as consequence, since the battery stays still, goes in increasing the chemical potential of its compounds; PbO2 for example has a greater chemical potential than PbO or PbSO4. Essentially the chemical potential is related with the electronic potential of the chemical species. Simplifying furthermore, the increase in mass goes in the increase of the electromagnetic fields of the chemical's electrons.A simple example: you give electromagnetic energy (of the right frequency) to an hydrogen's atom. The atom absorbs a photon and as a consequence its electron jumps up to the first excited level. At this moment the hydrogen atom has not the mass it had before: its mass is increased (almost exactly) of E/c2 where E is the photon's energy."Almost exactly" because the atom don't stay exactly still after have absorbed the photon (anyway the difference is very small).Where does the atom's mass increase has gone? It has gone in the increased electromagnetic field in the space between the nucleus and the atom's outer edge (the excited atom has a different spatial distribution of the electron).